Write the R code and test it.

1. Derive the maximum likelihood estimate for the regression vector (β) in the linear regression model (i.e., Equation 7.5).

2. Develop an MH algorithm for the linear regression model that samples all regression parameters simultaneously. How difficult is it to find a proposal density that yields a reasonable acceptance rate? What are some possible strategies for finding a good proposal density?

3. How can we work with the original posterior density (not the logged version) without worrying about overflow/underflow problems. Hint: Consider computing the ratio case by case. Write the R code and test it.

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